Classical results on the sample complexity of neural networks normally have strong dependency on the dimensions of parameter matrices. The upper bound shown in [1]} is based on VC dimension and strongly rely on both the depth and the width of the network. In [2]}, the authors make use of contraction principle and show an upper bound that have an exponential dependence on the network depth \(d\) , even if the norm of each parameter matrix is strictly upper bounded. Specifically, if input data \(\mathbf {x}_i, i\in [m]\)  satisfies \(||\mathbf {x}_i||\le B\)  and each parameter matrix \(W_j,j\in [d]\)  satisfies \(||W_j||_F\le M_F(j)\) , then under some assumptions on the activations, the results in [2]} show that
\(\hat{\mathcal {R}}_m*{\mathcal {F}*{X^m}} \sim \mathcal {O}*{\frac{ B2^d\prod _{j=1}^d M_F(j) }{ \sqrt{m}}},\) 
